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A085238 Sort the numbers 2^i and 3^j. Then a(n) is the exponent of the n-th term.

%I A085238 #22 Mar 25 2025 15:11:51
%S A085238 0,1,1,2,3,2,4,3,5,6,4,7,5,8,9,6,10,11,7,12,8,13,14,9,15,10,16,17,11,
%T A085238 18,19,12,20,13,21,22,14,23,15,24,25,16,26,17,27,28,18,29,30,19,31,20,
%U A085238 32,33,21,34,22,35,36,23,37,38,24,39,25,40,41,26,42,27,43,44,28
%N A085238 Sort the numbers 2^i and 3^j. Then a(n) is the exponent of the n-th term.
%H A085238 Amiram Eldar, <a href="/A085238/b085238.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..500 from T. D. Noe)
%F A085238 A006899(n) = A085239(n)^a(n).
%F A085238 a(A085240(n)) = a(n).
%t A085238 seq[lim_] := Module[{r2 = Range[0, Floor[Log2[lim]]], r3 = Range[0, Floor[Log[3, lim]]]}, Rest@ SortBy[Join[{#, 2^#} & /@ r2, {#, 3^#} & /@ r3], Last][[;; , 1]]]; seq[10^14] (* _Amiram Eldar_, Mar 25 2025 *)
%o A085238 (Haskell)
%o A085238 a085238 n = e (mod x 2 + 2) x where
%o A085238    x = a006899 n
%o A085238    e b p = if p == 1 then 0 else 1 + e b (p `div` b)
%o A085238 -- _Reinhard Zumkeller_, Oct 09 2013
%o A085238 (PARI) do(lim)=my(v=List(vector(logint(lim\=1,2),i,1<<i))); for(i=0,logint(lim,3), listput(v,3^i)); apply(n->my(t=valuation(n,2)); if(t, t, valuation(n,3)), Set(v)) \\ _Charles R Greathouse IV_, Sep 02 2015
%o A085238 (Python)
%o A085238 from sympy import integer_log
%o A085238 def A085238(n): return k+1 if 6**(k:=integer_log(m:=3**(n-1),6)[0])<<1<m else integer_log(1<<n,6)[0] # _Chai Wah Wu_, Feb 04 2025
%Y A085238 Cf. A006899, A085239, A085240.
%K A085238 nonn
%O A085238 1,4
%A A085238 _Reinhard Zumkeller_, Jun 22 2003